Minimal polynomial identities for right-symmetric algebras

Abstract

An algebra $A$ with multiplication $A\times A \to A, (a,b)\mapsto a\circ b$, is called right-symmetric, if $a\circ(b\circ c)-(a\circ b)\circ a\circ (c\circ b)-(a\circ c)\circ b,$ for any $a,b,c\in A$. The multiplication of right-symmetric Witt algebras $W_n=\{u\der_i: u\in U, U={\cal K}[x_1^{\pm 1},...,x_n^{\pm}$ or $={\cal K}[x_1,...,x_n], i=1,...,n\}, p=0,$ or $W_n({\bf m)}=\{u\der_i: u\in U, U=O_n({\bf m})\}$, are given by $u\der_i\circ v\der_j=v\der_j(u)\der_i.$ An analogue of the Amitsur-Levitzki theorem for right-symmetric Witt algebras is established. Right-symmetric Witt algebras of $ satisfy the standard right-symmetric identity of degree $2n+1:$ $\sum_{\sigma\in Sym_{2n}}sign(\sigma)a_{\sigma(1)}\circ(a_{\sigma(2)}\circ >...(a_{\sigma(2n)}\circ a_{2n+1})...)=0.$ The minimal deg$ left polynomial identities of $W_n^{rsym}, W_n^{+rsym}, p=0,$ i$ The minimal degree of multilinear left polynomial identity of $$ is also $2n+1.$ All left polynomial (also multilinear, if $p>0$) identities of right-symmetric Witt algebras of minimal $ combinations of left polynomials obtained from standard ones by permutations of arguments.